Anisotropic string tensions and inversely magnetic catalyzed deconfinement from a dynamical AdS/QCD model

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Physics Letters B

www.elsevier.com/locate/physletb

Anisotropic string tensions and inversely magnetic catalyzed deconfinement from a dynamical AdS/QCD model

Hardik Bohra

^a

, David Dudal

^b,c

, Ali Hajilou

^d,b

, Subhash Mahapatra

^a,∗

aDepartmentofPhysicsandAstronomy,NationalInstituteofTechnologyRourkela,Rourkela, 769008,India bKULeuvenCampusKortrijk–Kulak,DepartmentofPhysics,EtienneSabbelaan53bus7657,8500, Kortrijk,Belgium cGhentUniversity,DepartmentofPhysicsandAstronomy,Krijgslaan281-S9,9000, Gent,Belgium

dDepartmentofPhysics,ShahidBeheshtiUniversityG.C.,Evin,Tehran, 19839,Iran

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received9July2019

Receivedinrevisedform19December2019 Accepted20December2019

Availableonline27December2019 Editor:N.Lambert

We extend previous work on dynamical AdS/QCD models by introducing an extra ingredient under the form of abackground magnetic field,thisto gain insight intothe influence such field can have on crucial QCD observables. Therefore, we construct aclosed form analytic solution to an Einstein- Maxwell-dilaton systemwith amagneticfield.We specificallyfocus onthe deconfinementtransition, reportinginversemagneticcatalysis,andonthestringtension,reportingaweaker/strongerconfinement along/perpendiculartothemagneticfield.Thelatter,beingofimportancetopotentialmodellingofheavy quarkonia,isinqualitativeagreementwithlatticefindings.

©^{2019TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

Quantumchromodynamics (QCD) is the quantumfield theory ofstrong interactions capable of describing sub-atomics particles suchasquarksandgluons. Twoofitsmaincharacteristicfeatures are colorconfinement andchiralsymmetry breaking [1]. It is by nowwellknownthatQCDatlowtemperatureandchemicalpoten- tial exhibitsconfinement and chiral symmetry breaking whereas athightemperatureandchemical potential itundergoes a phase transitionto a chiral symmetry restoredphase where deconfine- mentalso sets in.Understanding thecomplete phase diagram of QCDintheparameterspaceoftemperature,chemicalpotentialetc.

isachallenging task,andisofutmostimportance inhighenergy physics. Indeed,the investigation ofthe QCD phase diagram and thesearchofnewphasesofmatterareofgreatrelevance,attract- ing worldwide attention, be it from the experimental, lattice or theoreticalcommunities[2].

Recent experiments withrelativistic heavy ion collisions have suggestedthepossibilityofnewparametersintheQCDphasedia- gram.Inparticular,itisexpectedthataverystrongmagneticfield e B∼⁰.3 GeV², whichiscreatedintheearly stagesofnoncentral relativisticheavy ioncollision[3–7], mighthaveimportantconse-

*

Correspondingauthor.

E-mailaddresses:[email protected](H. Bohra),

[email protected](D. Dudal),[email protected](A. Hajilou), [email protected](S. Mahapatra).

quences on the QCD phase diagram [8–10]. The extremelylarge magneticfieldrapidlydecaysafterthecollision,however,itisex- pectedthatitstillremainssufficientlylargeatthetime whenthe quark-gluon plasma (QGP)forms [11,12], henceit can affect the QCD matter nearthe deconfinement transition temperature[10].

ThisexpectationhasledtoanintenseinvestigationofQCDinthe presenceofa backgroundmagneticfield. Becauseofits manyin- teresting properties and phenomenological relevance for e.g. the chiralmagneticeffect[13,14],(inverse)magneticcatalysis[15–36], early universe physics [37,38], dense neutron stars [39] etc., the areaofmagnetisedQCDdrewmuchattentioninrecentyears.For detailedreviewsonthesesubjects,seeforexample[8,9].

It was expected from the work of [40,41] that the magnetic field has a constructive effect on the quark condensate andthe deconfinement transition temperature, a phenomenon commonly termedasthemagneticcatalysis.Furtherinvestigations,bothfrom lattice simulations and from theoretical models based on weak coupling approximations, had confirmed these results [15–24].

However, it came asa big surprisewhen state of the artlattice calculationsinsteadfoundinversemagneticcatalysis,i.e. themag- neticfieldwasfoundtofacilitatethedestructionofthequarkcon- densateandthencedecreasedthetransitiontemperature[25–27].

Subsequently,severalinterestingphysicalscenarios,thoughnoten- tirelysatisfactory,weresuggestedtoexplainthereasonbehindthe inversemagneticcatalysis,seeforexample[28–36].

It is widely expected that the inverse magnetic catalysis be- haviourmainlyresultsfromthestronglycoupleddynamicsaround https://doi.org/10.1016/j.physletb.2019.135184

0370-2693/©^{2019TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

thetransitiontemperature,asmostperturbativeQCD calculations and other effective models instead suggest magnetic catalysis.

Since the dominant physics near the deconfinement and chiral phase transition is non-perturbative, it is therefore more appro- priatetostudythemagneticfieldeffectsinQCDusingtechniques thataremorereliable atstrongcoupling.Heretheideaofgauge- gravity duality, apart from the usual lattice calculations, appears asanaturalcandidate[42–44].Theideathatcertainstronglycou- pled field theories without any gravitational degrees of freedom can be mappedto classical Einstein gravityvia the gauge-gravity dualityprovidesanelegantmethodbywhichthestronglycoupled regime ofQCD canbe probed.Indeedbynow, aftercertainmod- ificationsoftheoriginalmodelofthegauge-gravityduality,many characteristicfeaturesofQCDhavebeenreproducedfromhologra- phy,andinsome cases,newandinterestingpredictionshavealso beenmade,forarecentreviewsee[45].

In recent years, the effect ofa background magnetic field on thechiralcondensateanddeconfinementtransitionhasalsobeen discussed holographically. This question has been addressed in both top-down aswell asbottom-up models ofholographic QCD [46–64]. The top-down models, —unfortunately, not entirely ap- propriatetodescribeQCD-likephysicsinthefirstplace,butmore satisfactory as far as the correctness and validness of the du- ality is concerned—predict magnetic catalysis [46]. On the other hand, some phenomenological bottom-up holographic soft and hardwallQCDmodelsdopredictinversemagneticcatalysisforthe deconfinement transition, however, they continue to show mag- neticcatalysisbehaviourforthechiraltransition[51,55].Moreover, thesesoft andhard wall models donot always solve the gravity equationsexplicitlyandinmostcasestherunningofthecoupling constant (or the dual dilaton field) is introduced by hand in an adhocwayintotheEinstein-Hilbertaction.Inrecenttimes,more advanced phenomenological bottom-up holographic QCD models, whichcorrectlysolvethegravityequations,havebeenconstructed that displayinversemagnetic catalysis.In 2+1 dimensions, sen- sible gravity solutions displaying inverse magnetic catalysis have beenfoundin[57–59],while in3+^1,[61,62] displayedthepos- sibilityofinversecatalysisinthedeconfinementaswellasinthe chiralsector,dependingonthevalueofanewparametercwhich can influence AdS/QCD at vanishing magnetic field as well. The specific rôle and influence of this parameter c is to the best of ourknowledgeaninterestingopenquestion.

Anotherinteresting inherently non-perturbativeQCD quantity, with direct observable consequences, is the string tension be- tweenheavyquarks.Thisisrelevanttounderstandingthebinding (and consequent meltingathigher temperatures) of heavy quark statessuch ascharmonium,inparticular whenrelying on poten- tialmodelling [65,66]. Original lattice dataof [67,68] predictsan increase,respectivelydecrease,inthestringtensionperpendicular, respectively, parallel to the quark-antiquark orientation. Support- ing evidencefor such scenariocame in frommodellingthe non- perturbative QCD vacuumina specificway[69].In [70],a semi- classicalreasoningwasprovidedtoargueagainstQCDstringbreak- ing in the perpendicular directionfor sufficiently large magnetic field. To our knowledge, the magnetic field induced anisotropies inthestringtensionarenotthatwellexploredfromaholographic viewpoint(ageneraldiscussioncanbefoundin[71])andwewant tobridge thisgap.¹ Ofcourse,specific effectsthat anisotropycan haveinaholographiccontext,havebeenexplored,seee.g. [73–77].

1 Althoughtheanisotropiceffectsofthebackgroundmagneticfieldontheprobe quark-antiquarkfreeenergyhavebeendiscussedpreviouslyin[72],however,itdid notprovideanyconcreteresultonthestringtension,whichisnotsurprisinggiven thenon-confiningN=^{4 SYMsetupofthelatterwork.}

One of the main problems that have hindered the construc- tion ofa genuinephenomenologicalholographic QCDmodelwith a background magnetic field is the difficulty to find a dilaton backreacted magnetised AdS solution. This will require a simul- taneous solution of the Einstein-Maxwell-dilaton system with a non-trivial and consistent profile for the dilaton field. Indeed, a magneticfield embeddedEinstein-Maxwell-dilaton gravitysystem corresponds toafewsecond-order non-linearcoupleddifferential equations. Closed-form analytical solutions are rather difficult to be found.Inthiswork,usingthepotential reconstructionmethod [78–87],wewillremedythisproblemandfindacompletesolution to the Einstein-Maxwell-dilaton gravity system, containing both the magnetic field as well as the running dilaton. In particular, we will show thata consistent solutionto theEinstein-Maxwell- dilaton system can be found in terms of a single scale function A(z) (see eqs. (2.12), (2.13), (2.15), (2.16) and (2.17)). This scale functionwillbefurtherchosenbytakinginputsfromrealQCDand bymatchingholographicQCDresultswithrealQCDwithvanishing magneticfield.Moreover,inadditiontothefinitetemperatureand magnetic field,we extendourmodeltoincludethechemical po- tentialaswell.Thisisdesirableascomputationswithfinitechem- ical potential arecurrentlyvery challengingforlattice techniques duetothewell-knownsignprobleminEuclideanspace-time.

2. Einstein-Maxwell-dilatongravitywithamagneticfield InordertoconstructamagnetisedholographicQCDmodelwith runningdilatonandchemicalpotential,weconsiderafivedimen- sional Einstein-Maxwell-dilaton (EMD) gravity system with two Maxwellfields,

SE M

= −

¹ 16

π

G5

d⁵x

√

−

^g

R

−

^f1

(φ)

4 F₍1)M NF^{M N}

−

^f2

(φ)

4 F(2)M NF^{M N}

−

¹

2

∂

M

φ∂

^M

φ −

V

(φ)

,

(2.1)

where F_{(1)M N} and F_{(2)M N} are the field strength tensors for two U(1) gaugefields,φ is thedilatonfield, f1(φ) and f2(φ)are the gaugekineticfunctionsrepresentingthecouplingbetweenthetwo U(1)gaugefieldsononehandandthedilatonontheotherhand.

V(φ)isthepotentialofthedilatonfield,whoseexplicitformwill dependonthescalefunction A(z)(seebelow),andG5istheNew- ton constant in five dimensions. The inspiration forthis kind of modelling came from[84], albeit that the latter work happened inadifferentcontext.ConcerningtheinterpretationoftheAbelian gauge fields,wecan consider A₁ asthedualofa (neutral)flavor current, capable ofcreating mesons,while A2 isthe dual of the electromagneticcurrent.Inprinciple,thelattercancreateadiffer- ent neutralmeson. Sincewe workwith U(1)×^U(1),themesons arechargeneutral,sowecannotdirectlycoupleelectromagnetism to them. Indeed,we do not have a direct coupling between the 2 gauge fields, so we will never be able to couple e.g. a mag- netic field B to the neutral meson(s). For our current purposes, we willemploy thesecond gauge field just to introduce a (con- stant)magneticfield B,i.e. wehavenointerestinthefluctuations of A₂.Noticethatthis Bisthe5-dimensionalmagneticfield,that needs to be suitably rescaled via the AdS length L to get the physical,4-dimensional,magnetic field B. Howtodo thiscanbe found in[51]. As we aremostly interested in qualitativefeatures intermsofthemagneticfield,wewilldeliberatelykeepusingthe 5-dimensional B.

Onemajordrawbackoftheaction(2.1) isthatitneitherexplic- itlyincorporatesthedynamicsofthechiralcondensatenorthatit directlycouplesthechiralcondensatetothemagneticfield.Conse- quently,theeffectsofthemagneticfield onthechiralcondensate

can only be incorporated indirectly fromthe backgroundmetric, seealso[51].Therefore,contrarytorealQCD,thebackreactionof thechiralcondensateonthequark-antiquarkfreeenergyandvice versa willbe completely ignored in the holographic model (2.1).

Thisis a major disadvantage of all probe-approximatedAdS/QCD models,likeeq. (2.1),wherenoexplicitinterplaybetweenthechi- ralcondensateandPolyakovloopexists.Amoreaccurateandreal- isticholographicQCDmodel,thatincorporatesthebackreactionof thechiralfieldonthespacetimegeometryfromthestart,would—

although very interesting—be extremely challenging to construct analytically.Wearethusworkingwithakindofholographicana- logueofthequenchedQCDapproximationknownfromlatticeQCD (nodynamical quarks).As such, onemightquestion how toeven couplethemagneticfield tothetheory ifthereare nodynamical chargecarriers.Here,wefollowthepragmaticapproachofe.g. [88, 89],initselfmagneticfield-dependent generalizationsofthesem- inalworks[90,91],andwethusconsiderourengineeredboundary model capable of mimicking some essential QCD features, after whichitcanbeusedtodescribe,withoutfurtherinput,otherQCD- ish properties. Notice that Einstein-Maxwell-dilaton models have been used throughout literature as an effective way to describe QCDinpresenceofelectromagneticbackgroundfields,asitisevi- dentfromourextensivereferencelist.

Byvaryingtheaction(2.1) onecanderivetheequationsofmo- tionforEinstein,Maxwellanddilatonfields.Since,inthisworkwe are mostly interested in a magnetised black brane solution with runningdilaton,weconsiderthefollowingAnsätze forthemetric gM N,fieldstrengthtensor F₍i)M Nanddilatonfieldφ,

ds²

=

^L2S

(

z

)

z²

−

^g

(

z

)

dt²

+

^dz2

g

(

z

) +

^dy2₁

+

^eB2z2

dy²₂

+

^dy2₃

, φ = φ (

z

),

A₍1)M

=

At

(

z

)δ

^t_M

,

F₍2)M N

=

Bdy2

∧

dy3

,

(2.2) where S(z) isthescalefactor, L istheAdS lengthscaleand g(z) istheblackeningfunction.zistheusualholographicradialcoordi- nate,andinourcoordinatesystemitrunsfromz=0 (asymptotic boundary)toz=^zh (horizonradius),ortoz= ∞^{forthermalAdS} (without horizon).We introduced abackground magneticfield B inthe y1-direction.Becauseofthisbackgroundmagneticfield,the systemnolongerenjoys theS O(3)invarianceinboundaryspatial coordinates (y₁, y₂, y₃), and we precisely chose the metric An- sätze such that assoon as we switch offthe magnetic field the S O(3)invarianceisrecovered.

UsingtheAnsätzeofeq. (2.2) wegetfourEinsteinequationsof motion,

g

(

z

) +

^g

(

z

)

2B²z

+

^3S

(

z

)

2S

(

z

) −

³

z

−

^z2f1

(

z

)

A_t

(

z

)

²

L²S

(

z

) =

⁰

.

(2.3) B²ze^−2B2z2f₂

(

z

)

L²S

(

z

) +

^2B2g

(

z

) +

g

(

z

)

4B⁴z

+

^3B2S

(

z

)

S

(

z

) −

^4B2

z

=

0

.

(2.4)

S

(

z

) −

^3S

(

z

)

²

2S

(

z

) +

^2S

(

z

)

z

+

^S

(

z

)

4B⁴z² 3

+

^4B2

3

+

¹ 3

φ

(

z

)

²

=

⁰

.

(2.5)

g

(

z

)

3g

(

z

) +

^S

(

z

)

S

(

z

) +

^S

(

z

)

7B²z

2S

(

z

) +

^3g

(

z

)

2g

(

z

)

S

(

z

) −

⁶

zS

(

z

)

+

^g

(

z

)

5B²z

3g

(

z

) −

³ zg

(

z

)

+

^2B4z2

+

^B2z2e−2B

2z²f2

(

z

)

6L²g

(

z

)

S

(

z

) −

^6B2

+

^2L2S

(

z

)

V

(

z

)

3z²g

(

z

) +

^S

(

z

)

² 2S

(

z

)

²

+

⁸

z²

=

0

.

(2.6)

Similarlywe getthe followingequation ofmotionforthe dilaton field,

φ

(

z

) + φ

(

z

)

2B²z

+

^g

(

z

)

g

(

z

) +

^3S

(

z

)

2S

(

z

) −

³

z

+

^z2At

(

z

)

² 2L²g

(

z

)

S

(

z

)

∂

f₁

(φ)

∂φ

−

^B2z2e−2B

2z²

2L²g

(

z

)

S

(

z

)

∂

f₂

(φ)

∂φ −

^L2S

(

z

)

z²g

(

z

)

∂

V

(φ)

∂φ =

⁰

,

(2.7)

andtheequationofmotionforthefirstgaugefield, A_t

(

z

) +

^A_t

(

z

)

2B²z

+

^f1

(

z

)

f₁

(

z

) +

^S

(

z

)

2S

(

z

) −

¹ z

=

⁰

.

(2.8)

Onecanexplicitlycheckthattheequationofmotionforthesecond Maxwell field is trivially satisfied andhence it willnot give any additional equation. Therefore, we have in total six equations of motion. However, only five of them independent. Below we will choose the dilaton equation (2.7) as a constrained equation and considertherestoftheequationsasindependent.Inordertosolve thelatter,weimposethefollowingboundaryconditions,

g

(

0

) =

^{1 and g}

(

z_h

) =

⁰

,

A_t

(

0

) = μ

and A_t

(

z_h

) =

⁰

,

S

(

0

) =

¹

,

φ (

0

) =

⁰

,

(2.9)

where

μ

isthechemicalpotentialoftheboundarytheorywhichis relatedto thenearboundaryexpansion ofthezeroth component ofthefirstgaugefieldand,asmentionedbefore,zh isthelocation ofthe blackholehorizon. Apartfromtheseboundary conditions, we will also assume that the dilatonfield φ remains real every- whereinthebulk.Aswewillseelater,thisconditionwillseverely restrictouranalyticsolutionforafinitemagneticfield.

Inordertosolveeqs. (2.3),(2.4),(2.5),(2.6) and(2.8) simulta- neously,weadoptthefollowingstrategy.

1. We first solve eq. (2.8) and obtain the solution for A_t(z) in termsof f1(z)andS(z).

2. Using At(z),wethen solveeq. (2.3) andfindthesolutionfor g(z)intermsof f₁(z)andS(z).

3. Using g(z),wethensolveeq. (2.4) toobtain f2(z). 4. Next,wesolveeq. (2.5) andfindφ(z)intermsofS(z). 5. Finally, we solve eq. (2.6) and obtain the dilatonpotential in

termsof S(z)andg(z).

Applying theabove strategyandsolvingeq. (2.8),we getthefol- lowingsolutionforA_t

A_t

(

z

) =

^K1

z 0

d

ξ ξ

e^−B2ξ2 f1

(ξ ) √

S

(ξ ) +

^K2

.

(2.10)

Applyingtheboundarycondition(eq. (2.9)),weget K2

= μ ,

K1

= − μ

_zh

0 d

ξ

^ξe−B2ξ2

f1(ξ )√ S(ξ )

,

(2.11)

andthesolutionforAt thenbecomes,

A_t

(

z

) = μ

1

−

z

0 d

ξ

^ξe−B

2ξ² f1(ξ )√

S(ξ )

_zh

0 d

ξ

^ξe−B2ξ2

f1(ξ )√ S(ξ )

= ˜ μ

z_h

z

d

ξ ξ

e^−B2ξ2 f₁

(ξ ) √

S

(ξ ) .

(2.12) Substitutingeq. (2.12) intoeq. (2.3),wegetthefollowingsolution forg(z),

g

(

z

) =

z 0

d

ξ ξ

³e^−B2ξ2

S³

(ξ )

K3

+ μ ˜

² L²

ξ 0

d

ξ ˜ ξ ˜

e^−B2ξ˜2 f₁

( ξ ) ˜

S

( ξ ) ˜

+

K4

.

(2.13)

TheconstantsK3andK4canbefixedfromeq. (2.9) andweget K₄

=

¹

,

K3

= −

¹

_zh

0 d

ξ

^ξ3e−B

2ξ² S³(ξ )

×

1

+ μ ˜

² L²

z_h

0

d

ξ ξ

³e^−B2ξ2

S³

(ξ )

^ξ

0

d

ξ ˜ ξ ˜

e^−B2ξ˜2 f₁

( ξ ) ˜

S

( ξ ) ˜

.

(2.14)

Thecouplingfunctioncanbeobtainedfromeq. (2.4),

f2

(

z

) = −

^e2B

2z²L²S

(

z

)

z

g

(

z

)

4B²z

+

^3S

(

z

)

S

(

z

) −

⁴

z

+

^2g

(

z

)

.

(2.15)

Similarly,thedilatonfieldcanbeobtainedfromeq. (2.5)

φ

(

z

) =

^{−8B4z3S(z)2−8B2zS(z)2−}√^{6zS(z)S(z)+9zS(z)2−12S(z)S(z)}

2zS(z)

,

φ (

z

)

=

dz

−8B⁴z³S(z)²−8B²zS(z)²−6zS(z)S(z)+9zS(z)²−12S(z)S(z)

√2zS(z)

+

K5 (2.16)

wheretheconstant K5 will befixed demandingthat² φ|z=⁰→^0.

Andfinally,eq. (2.6) allowsustofindthepotential, V

(

z

) =

^g

(

z

)

L²

−

^9B2z3S

(

z

)

2S

(

z

)

²

+

^10B2z2

S

(

z

) −

^3z2S

(

z

)

² S

(

z

)

³

+

^12zS

(

z

)

S

(

z

)

²

+

^z2

φ

(

z

)

² 2S

(

z

) −

¹²

S

(

z

)

−

^z4f1

(

z

)

A_t

(

z

)

² 2L⁴S

(

z

)

²

+

^g

(

z

)

L²

−

^B2z3

S

(

z

) −

^3z2S

(

z

)

2S

(

z

)

²

+

^3z

S

(

z

)

.

(2.17) It isclear fromtheabove equations that acomplete analytic so- lutionto theEinstein-Maxwell-dilaton system withabackground magneticfieldcanbeobtainedintermsoftwoarbitraryfunctions, i.e. the scale function S(z) and the gauge coupling f1(z). Differ- entformsof S(z) and f₁(z)willgive differentphysicallyallowed solutions. Indeed, it can be explicitly verified that the Einstein, Maxwell anddilaton equationsare satisfiedforany formof S(z) and f₁(z). Thus we havefound a familyof analytic solutions for

2 ThissimplechoiceassuresweasymptotebacktoAdS5neartheUVQCDbound- aryz=^0.

the gravity system of eq. (2.1). Since our aim here is to model real QCD properties holographically, we will fix these two arbi- trary functions by taking inputs fromreal QCD. For example, by comparing the holographic results for the deconfinement transi- tiontemperatureandmesonmassspectrumwithlatticeQCD,we canfix/constraintheformsofS(z)and f1(z).

Theformofthegaugecouplingfunction f1canbeconstrained bystudyingthevectormesonmassspectrum.Inparticular,bytak- ingthefollowingsimpleformof f₁,

f₁

(

z

) =

^e−cz

2−B²z²

√

S

(

z

) ,

(2.18)

thevectormesonspectracanbeshowntolieonlinearReggetra- jectories for B=^0.In particular, the mass squaredof thevector mesonssatisfiesm_n²=^{4cn.Moreover, theparameterc canalsobe} fixed bymatchingwithlowestlyingheavymesonstates J/and ,andby doingthatwegetc=¹.16 GeV²,see[78,83] formore details.

Substitutingeq. (2.18) intoeqs. (2.13),(2.15) and(2.16),andus- ing S(z)=^{e2A(z),wegetthefollowingsolutions,}

g

(

z

) =

¹

+

z 0

d

ξ ξ

³e^{−B2ξ2−3A(ξ )}

K3

+ μ ˜

² 2cL²e^cξ2

,

with K3

= −

1

+

_2cL^μ˜22

_zh

0 d

ξ ξ

³e^{−B2ξ2−3A(ξ )+cξ2}

_zh

0 d

ξ ξ

³e^{−B2ξ2−3A(ξ )}

.

(2.19) f2

(

z

) = −

^L2e2B

2z²+^2A(z) z

g

(

z

)

4B²z

+

6A

(

z

) −

⁴ z

+

2g

(

z

)

.

(2.20)

φ (

z

)

=

dz

−

² z

3z A

(

z

) −

^{3z A}

(

z

)

²

+

^6A

(

z

) +

^2B4z3

+

^2B2z

+

^K5

.

(2.21)

Notice that √

S(z), appearing in eq. (2.18), is then well-defined.

Also, one can substitute eq. (2.18) into eqs. (2.12) and (2.17) to obtain the explicit solutions for At(z) and V(z). Therefore, in eqs. (2.12),(2.13), (2.15),(2.16) and(2.17) acompletesolutionfor the Einstein-Maxwell-dilaton gravity system is obtainedin terms ofasinglescalefunction A(z).

Letusalsonotetheexpressionsofblackholetemperatureand entropy, which willbe usefullater on inthe investigation ofthe blackholethermodynamics

T

= −

^z

3

he^{−3A(zh)−B2z2h} 4

π

K3

+ μ ˜

² 2cL²e^cz2h

,

S

=

^eB

2z_h²+3A(zh)

4z³_h

.

(2.22)

Beforeweclosethissection,itisimportanttoemphasizeagain thateqs. (2.17)-(2.21) areasolutionoftheaction(2.1) foranyscale factor A(z).We thereforehavean infinitefamilyofanalytic black hole solutions for thegravity system ofeq. (2.1). Thesedifferent solutions howevercorrespond to differentdilatonpotentials (and thereforetodifferentactions),asdifferentformsof A(z) willgive differentpotentials V(z).However, oncetheformof A(z)isfixed

thentheformofV(z)alsois,andinreturneqs. (2.17)-(2.21) cor- respond to a self-consistent solution to a particular action with predeterminedA(z)andV(z).

OnemightalsoworrythatthedependenceofV ontheparame- terszh,

μ

andBistroublesomeasitindicatesthatdifferentvalues of theseparameters correspond to a different action, andthere- fore,toadifferentgravitymodelaltogether.Weliketoemphasise here that V does not depend on these parameters explicitly at thelevel ofthe action orequations ofmotion.These parameters appear only when the boundary conditions in eq. (2.9) are im- posed.Indeed,noticethat thereisasecond independentsolution toour EMD equations ofmotion, whichcorresponds tothermal- AdS. For the thermal-AdS, we have g(z)=^{1. Then it is easy to} inferfromeq. (2.6) thatV isindependentofzh and

μ

(the B de- pendenceappearsbecauseofthemetricandgauge fieldansätze).

Ifthepotentialweredependentonz_hand

μ

fromthebeginningin theactionitself,thenthepotentialinthethermal-AdSbackground wouldhavedependedonz_h and

μ

aswell,whichiscertainlynot thecaseinourmodelaswehavealludedtoabove.Therefore,one shouldinterpretthedependenceofthepotential V onzh and

μ

as anon-shell dependence,andnot asan off-shellone.In anycase, we have numerically checked that the on-shell V depends only verymildlyonzh,

μ

andB.Inparticular,thepotentialprofilesfor differentz_h,

μ

andBvaluesarealmostindistinguishablefromeach otherintheregionawayfromthehorizonwhereastheyaresepa- rableonlymildlyinthenearhorizonregion.Illustrativefiguresare includedinAppendixA.

3. Results

Following[78],wewilldepartfromtheB=^{0 Ansatz3}

A

(

z

) = −

^az2

.

(3.1)

Letusfirstnotetheexpressionofφ (z)

φ (

z

) =

^9a−B2log

√

6a²−^B4√

6a²z²+^9a−^B4z2−^B2+^6a2z−^B4z

√6a²−^B4

+

^{z 6a2z2}

+

^9a

−

^B2

B²z²

+

¹

−

9a

−

^B2

log

√

9a

−

^B2

√

6a²

−

^B4

√

6a²

−

^B4

.

(3.2)

Similar expressions can be found for At(z), g(z), f2(z) and V(z) aswell.However, theseexpressions are too lengthy toreproduce hereandalsonotparticularlyilluminating,wethereforejustmen- tionedthefunctionalityofφ (z)sinceitgivesthestability criteria ofour solution.Indeed, fromeq. (3.2), we learn that the dilaton field is real-valued only when B⁴≤^{6a2. This condition severely} restricts the validity of our gravity solution and below we will workwithonlythose valuesofa and B forwhich thiscondition issatisfied.Moreover,itisalsointerestingtonotethattheGubser criterion [94]—the scalarpotential must be bounded fromabove, V(z)≤^V(0), for a physically acceptable holographic solution—is always satisfied under the same condition B⁴≤^{6a2. In partic-} ular, the potential is almost constant having value −¹²/L² near theasymptoticboundaryandthendecreases inthedeepIR.This, therefore,givesastrongself-consistencycheckonourconstructed solution.

3 Inongoingwork,amoregeneralformfactorwillbeemployedsothatnextto aconfininglinearpotential,alsoasymptoticfreedomcanbebuiltin,seee.g. [92, 93].Inanycase,ourmain resultsconcerningtheanisotropicstring tensionwill remainthesameevenafteremployingamoresophisticatedformfactorwhichwill guaranteeasymptoticfreedominUV.

Fig. 1.TemperatureT asafunctionofhorizonradiuszhforvariousvaluesofthe magneticfieldBandμ=0.Herered,green,blue,brownandorangecurvescorre- spondtoB=^0,0.15,0.30,0.45 and0.6 respectively.InunitsGeV.

Fig. 2.FreeenergyFasafunctionoftemperatureTforvariousvaluesofthemag- neticfieldBandμ=^{0.Herered,green,blue,brownandorangecurvescorrespond} toB=^0,0.15,0.30,0.45 and0.6 respectively.InunitsGeV.

3.1. Blackholethermodynamicsandconfinement-deconfinementphase transition

Thethermodynamicsofthegravitysolutionwiththescalefac- tor ofeq. (3.1) is showninFigs. 1 and2.In Fig.1, thevariation ofHawking temperature T withrespectto the horizonradius zh forvariousvaluesofthemagneticfield B isshown. Wefindthat there exists a minimumtemperature Tmin belowwhich no black holesolutionexist.However,forT>Tmin,therearetwoblackhole solutions,a largeandasmallone, whicharemarked by 1 and 2 respectively.ThesmallblackholephaseforwhichT increases withzh isunstablewhereas thelargeblackholephase forwhich T decreases with z_h is stable. The unstable-stable nature of the small-largeblackholephasescanbeseenfromthefreeenergybe- haviourshowninFig.2.Here,wehavenormalisedthefreeenergy ofthe blackholewithrespect tothethermal AdScase, zh→ ∞^. Weseethatthefreeenergyofthesmallblackholephaseisalways largerthan thelarge blackholeandthermalAdSphases, indicat- ingtheunstablenatureofthissmallblackholephase.Importantly, upon varying the Hawking temperature, a phase transition from the large black hole phase to thermal AdS phase takes place at acriticaltemperature T_crit.Thisisthefamousblackhole-thermal AdSHawking-Pagephasetransition[95].

Interestingly, the above thermodynamic behaviour occurs for small but finite values of the magnetic field as well. For finite magneticfield,weagainfindtheunstablesmall-stablelargeblack hole phases, with thermal AdS dominating the physics at small temperatures. Importantly, the Hawking-Page thermal AdS–black hole phase transition persists even for finite values of the mag- netic field. The main difference appears in themagnitude ofthe critical temperature Tcrit. In particular, Tcrit decreases for higher values ofmagnetic field. Thedependence of Tcrit on B is shown inFig.3,whichisalsooneofthemainresultsofthispaper.Since

Fig. 3.ThevariationofthermalAdS–blackholephasetransitioncriticaltemperature TcritwithmagneticfieldB.Hereμ=^{0 is}considered.InunitsGeV.

Fig. 4.ThevariationofthermalAdS–blackholephasetransitioncriticaltemperature Tcrit withmagneticfieldB forvariousvaluesofchemical potentialμ.Herered, green,blue,brownandorangecurvescorrespondstoμ=0.0,0.3,0.6,0.9 and1.2 respectively.InunitsGeV.

thesethermalAdSandblackholephasesintheusuallanguageof gauge-gravity dualityaredualto theconfinementanddeconfine- mentphasesinthedualboundarytheory,accordingly,ourresultin Fig.3predictsinversemagneticcatalysisforthedualconfinement- deconfinementtransition.OurresultinFig.3,therefore,providesa majorimprovementonseveralsoftandhardwallmodelsofholo- graphicQCD,whichdidalreadysuggestinversemagneticcatalysis inthedeconfinementsector,howeverbasedonaadhocchoiceof thedilaton.Here,we haveexplicitlyincluded thebackreactionof thedilatonfieldinanotoverlycomplicatedway.

One can naively expect by looking at Figs. 1 and 2 that the above scenario for the thermal AdS–black hole phase transition might changefor large magnetic fields.However, we need to be carefulhere.Asmentionedbefore,thedilatonfield,andtherefore ourgravity solution, only makes sense when thecondition B⁴<

B⁴_c =^{6a2 is satisfied.Since we took a}=⁰.15 GeV² for a decent matchwiththe lattice QCDdeconfinement temperatureat B=⁰ [78], our gravity solution is trustworthy only for Bc⁰.61 GeV.

Thenwefindthatforall B≤^Bc,thethermalAdS–blackholephase transitionoccurs.Wehavealsoexplicitlycheckedthatitpersistsin termsof avarying a.Moreover, thecritical temperatureT_crit de- creases with magnetic field even forthese different values of a, indicatingthe inversemagneticcatalysisagain,inlinewithinde- pendentlatticeQCDpredictions.

Wenow move onto discussthethermodynamic behaviour in thepresence ofchemical potential

μ

.We findsimilar results(as discussedabove) withfinite

μ

aswell, andtherefore,we can be verybrief here.Our resultsare summarizedinFig.4. Oneofthe main outcomes hereis that our holographic model continues to exhibit inverse magnetic catalysis behaviour for non-zero values of

μ

as well. Unfortunately, we do not have lattice results for (inverse)magnetic catalysisatfinite

μ

yet,aslattice simulations usuallysufferfromthesignproblemwith

μ

.Therefore,theresults

in Fig. 4can be considered asa genuine predictionof our holo- graphicmodelofeq. (2.1).

Moreover,wealsofind,likemanyotherholographicQCDmod- els,adecreasingpatternforthecriticaltemperaturewithchemical potential.

3.2. TheanisotropicQCDstringtension

In order to study the QCD string tension,our approach is to considerthefreeenergyF ^ofaq,^q¯ ^{pairviathedualofthegauge} invariant quantity fromwhich the q,^q¯ interaction energy can be extracted [96], i.e. via the holographic realization of the Wilson loop [97–100]. In particular, the gauge-gravity duality provides a correspondence between the F ^{of the q},^q¯ ^{pair with separation} distance thatevolvesover a largetime T andthe Nambu-Goto (NG)on-shellaction.⁴Thisactiondescribesthephysicsoftheopen string that evolves in time and sweeps out a two dimensional world-sheetwhich isboundedon theAdS boundaryby the rect- angularWilsonloop×^T.So,wehave

F

(,

T

) =

T S^on_{N G}^−shell

(,

T

) ,

(3.3)

where S_{N G}

=

¹

2

π

²_s

d

τ

d

σ −

^detGs

.

(3.4)

Here, Ts= _2π¹2 s

istheopen stringtension,thecoordinates(

τ

,

σ

) are used to parameterize the two-dimensional world-sheet and (Gs)αβ=(gs)M N∂αX^M∂βX^N, where X^M(

τ

,

σ

) indicates the em- bedding of the open string in the gravity background, g_s is the background metricinthe stringframe,⁵ asappropriate to extract the string(q,q)¯ free energy [45,89]. Gs is theinduced metric on thetwo-dimensionalworld-sheet.

Theabovemetricsolution(2.2) isintheEinsteinframe,andthe standardmethodtopasstothestringframecanbeobtainedfrom thedilatontransformation[78,89],i.e.(gs)M N=^{e 23φg}M N.So,the metricsolution(2.2) inthestringframeis,

ds²_s

=

^L2e2As(z) z²

−

g

(

z

)

dt²

+

^dz2

g

(

z

) +

dy²₁

+

e^B2z2

dy²₂

+

dy²₃

,

(3.5) where A_s(z)=^A(z)+ ¹₆ φ (z), with A(z) andφ (z) asgiven in eqs. (3.1) and(3.2),respectively.Inthefollowing,weconsidertwo casestoinvestigatetheeffectofmagneticfield ontheQCDstring tension,theparallelcase,i.e. whentheq,^q¯ ^{pairisorientedparallel} tothemagneticfield,andthenalsoaperpendicularorientation.

3.2.1. Parallelcase

In this case, to parameterize the string world-sheet, we use the static gauge, i.e.

τ

=^{t and}

σ

=^y1. So, one can obtain both connected anddisconnectedsolutions thatminimize the Nambu- Goto actionfromeqs. (3.4) and (3.5).Theconnectedsolutionis a

4 Inprinciple,becauseofthenon-trivialdilatonprofile,therecanbeanadditional termintheNGstringworldsheetactionthatdescribesthecouplingbetweenthe dilatonfieldandtwo-dimensionalRicciscalaroftheworldsheet.However,inthe large’tHooftlimitλ→ ∞^{—withwhichoneisimplicitlyalwaysworkinginthe} gauge-gravitycorrespondence—thistermwillbenegligiblebeing anO(α)∼^√1_λ contribution,andthereforeitisalwaysomittedinanykindofappliedgauge/gravity computation.

5 Hereafter,weusethesubscript“s”toindicatethatweareworkinginthestring frame.